Critical curves of a piecewise linear map

TL;DR

This paper investigates the parameter space of a piecewise linear planar map, focusing on the algebraic and geometric properties of critical curves where orbits recur to the boundary, linking symbolic dynamics with these curves.

Contribution

It characterizes the algebraic and geometric structure of critical curves in the parameter space of a piecewise linear map, connecting symbolic dynamics with boundary recurrence.

Findings

Critical curves are algebraic and geometrically structured.

These curves are determined by symbolic dynamics of boundary itineraries.

The study reveals relationships between dynamics and algebraic properties of the curves.

Abstract

We study the parameter space of a family of planar maps, which are linear on each of the right and left half-planes. We consider the set of parameters for which every orbit recurs to the boundary between half-planes. These parameters consist of algebraic curves, determined by the symbolic dynamics of the itinerary that connects boundary points. We study the algebraic and geometrical properties of these curves, in relation with such a symbolic dynamics.